Why use continuity at b to prove uniform continuity on a union of intervals

[Stephen Tashi]

Am fully convinced now. Thanks Stephen and Lavinia. g on [a, b] is continuous, so there is left hand continuity at b. Similarly h has Right hand continuity at b. We exploit this property. Once again, am I on right track?

I think you are on the right track, but things would be clearer if you present a self-contained statement of what is to be proven as well as the proof. The text you quote from Spivak refers to "statements (i) and (ii)" without showing what they are.

 

[Alpharup]

You are right. I should have posted it.
Here is definition of uniform continuity.

 

[Alpharup]

Also about (i) and (ii)
)

 

[Alpharup]